# Syntonic comma

Diatonic intervals
Prime
second
third
fourth
fifth
sixth
seventh
octave
none
decime
undezime
duodecime
tredezime
semitone / whole tone
Special intervals
Microinterval
Comma
Diësis
Limma
Apotome
Ditone Tritone
Wolf
fifth
Natural septime
units
Cent
Millioctave
Octave
Savart

The syntonic comma , also didymic comma (after Didymos the musician ), is a small musical interval in music theory , which corresponds to a frequency ratio of 81:80 (approx. 21.51 cents ). The interval plays a role in connection with mood systems and can be obtained by stringing together pure intervals .

## Size and frequency ratio

The syntonic comma is the difference between the Pythagorean third (frequency ratio 81 / 64 ) and the pure third ( 5 / 4 ). A Pythagorean third case corresponds to a ditone two Pythagorean whole tones ( 9 / 8 ):

syntonic comma = Pythagorean third - pure third = 2 Pythagorean whole tones - pure third 21.51 cents .${\ displaystyle \ approx}$

Since the frequency ratios are multiplied or divided when adding or subtracting intervals, the frequency ratio of the syntonic comma is calculated as follows:

${\ displaystyle \ textstyle {\ left ({\ frac {9} {8}} \ right) ^ {2} / {\ frac {5} {4}} = {\ frac {81} {64}} / { \ frac {5} {4}} = {\ frac {81} {80}} = 1 {,} 0125}}$

The syntonic comma also occurs as the difference between a large whole tone ( 9 / 8 ) and a small whole tone ( 10 / 9 to):

syntonic comma = large whole tone - small whole tone

Its frequency ratio is calculated as follows:

${\ displaystyle \ textstyle {{\ frac {9} {8}} / {\ frac {10} {9}} = {\ frac {81} {80}}}}$
Syntonic comma as the difference between a third shifted by 2 octaves and 4 fifths
Mid-tone tuning: the syntonic comma is tempered by reducing the fifths
Equal tuning: fifths and thirds are tempered to whole-numbered semitones, which means that the syntonic comma is omitted

Another design option is the difference of perfect fifths 4 ( 3 / 2 ) to a pure third ( 5 / 4 ) and 2 octaves ( 2 / 1 ):

syntonic comma = 4 perfect fifths - 2 octaves - pure major third
${\ displaystyle \ textstyle {\ left ({\ frac {3} {2}} \ right) ^ {4} / \ left (2 ^ {2} \ cdot {\ frac {5} {4}} \ right) = {\ frac {81} {80}}}}$

This relationship is the basis of the quarter-point meantone , wherein the fifths to 1 / 4 of the syntonic commas are reduced, with the aim that of 4-tempered fifths each a pure third results.

## example

The syntonic comma occurs with modulations when the intonation is pure . For example, when modulating from C major to G major, not only  f in  f sharp ( change of sign ), but also a with the frequency 440  Hz increases by a syntonic comma in an  a with 445.5 Hz (frequency ratio 445.5440 = 8180 ).

Modulation from C major to G major

 Soprano only

Because this change cannot be made with keyboard instruments , they are medium-tone , well-tempered or - nowadays almost consistently - tuned equally (see Pythagorean comma ).